Abstract
Following the Kubo-Ando theory of operator means we consider the weighted geometric mean of upper triangular matrices A and B whose main diagonals are all 1, named the upper unipotent matrices. We also present its binomial expansion Showing that the weighted geometric mean is a geodesic of symmetry in the symmetric space equipped with point reflection, known as the Loos symmetric space, we derive several binomial identities on the Lie group of upper unipotent (resp. the Lie algebra of nilpotent) matrices.
Acknowledgments
The authors are grateful to Professor Bruno Iannazzo for suggesting a formula for of upper unipotent matrices. All authors equally contributed to this work. No potential competing interest was reported by the authors.
Disclosure statement
No potential conflict of interest was reported by the author(s).